Area of a Triangle via the Cross Product:

P(2, 3, -1), Q(-1, 2, 3), R(3, 1, -2).

The area of the parallelogram with sides PQ and PR is equal to the magnitude of the cross product of vectors representing two adjacent sides:

Area(parallelogram) = |PQ ´ PR|. The area of the triangle is half of this.

First we need to determine the vectors:

PQ = [-1 - 2]i + [2 - 3]j + [3 - (-1)]k = -3i - j + 4k.

PR = [3 - 2]i + [1 - 3]j + [-2 - (-1)]k = i - 2j - k.

Hence the cross product is

Hence the area of the triangle is